I have found WSockets very easy to use.
http://www.delphi32.com/vcl/2142/

There should be simple example provided, but the components themselves are pretty straight forward.

(Recommend using TCP)
Set properties port and host for client and then call Open. onData event activates when data is sent. Also for simple program as peer to peer with small data moving i recommend using functions that send and receive string, for easiest possible experience for a starter.

On more advanced note, there is also a onError event with parameters explaining everything. Have some TMemo up to print out those errors for debugging, do client.close when server times out or make timer reconnecting etc... but it's optional.